Internal Necking Research Articles

On the anisotropic coalescence of elliptic cylindrical voids considering the geometric and distributive properties

The geometric and distributive properties of voids significantly influence anisotropic coalescence behaviour. However, this problem has received little attention owing to the complexity of considering all the properties in the current analytical framework of limit analysis. To address this issue, this study proposes an analytical framework based on an elliptic coordinate system, including the determination of the ligament zone, characterization of plastic flow, and derivation of the void coalescence criterion, for porous materials with various geometric and distributive properties, including size, shape, spacing, and orientation. This framework is motivated by our observations that the evolution of the void geometry and surrounding plastic flow can be well characterized by the grid of the elliptic coordinate system. Subsequently, an analytical function is proposed to determine the ligament zone and coalescence direction with various void properties. A hollow nonaxisymmetric cylindrical unit cell is proposed to describe this ligament zone, and the corresponding trial velocity field is derived by extending the previous Gurson-like velocity field into the elliptic cylindrical coordinate system. The rationality of the field is validated by comparing its equivalent strain rate field with numerical simulations. Finally, a coalescence criterion is derived via the limit analysis of the proposed unit cell undergoing internal necking. Two heuristic adjustments are formulated for the overflow phenomenon in the rigid zone and outer ligament zones. Numerical assessments with various void properties confirm the accuracy of the analytical model. The coalescence criterion can predict the independent and coupling effects of geometric and distributive properties on anisotropic void coalescence. This study provides possible solutions to future plasticity problems of ellipsoidal inclusions.

Direct FE2 multiscale modeling of hydrogen-induced cracking in reactor pressure vessels

Leveraging on a recently developed multiscale simulation method, we propose a pioneering strategy for the multiscale analysis of hydrogen-induced cracking in reactor pressure vessels that considers micro-voids, their influences on hydrogen diffusion as well as the operating pressure of the vessel. In the finite element analysis, the nodal lattice hydrogen concentration and displacements are exchanged bi-directionally between the macroscale and microscale within a monolithic solution procedure. During the analysis, the voids are shown to cause non-uniform lattice hydrogen diffusion and the trapping effect, resulting in inter-void hydrogen accumulation, is demonstrated. Based on the hydrogen concentration on the void surfaces, the void hydrogen pressure was calculated for the hydrogen-induced cracking analysis. It was found that both internal shearing and necking failure modes exhibited micro-structure dependence. The internal shearing mode caused oriented through-wall propagation of cracks in the vessel pressure while the internal necking mode promoted void nucleation in the reactor pressure vessel. As a result, during the early stages of void evolution, the oriented internal shearing of the dispersed small voids showed a high possibility to induce through-wall failure of the reactor pressure vessel.

Yield criterion for intergranular void coalescence under combined tension and shear

Intergranular ductile fracture is a failure mode that may arise in many metallic alloys used in industrial applications. It manifests as the successive nucleation, growth, and coalescence of cavities at grain boundaries. Thus, simulation of intergranular ductile fracture in polycrystals requires modeling those three different stages at the scale of grain boundaries, i.e. at the interface between two different crystals. In this study, a yield criterion for the coalescence of cavities at the interface between two isotropic materials obeying Mises plasticity is first developed by limit analysis in order to provide some insights into that phenomenon. This criterion is checked against numerical limit analysis under combined tension and shear and is found to agree with unit-cell simulations quantitatively. The model is then extended to crystals so as to account for the complex coupling between loading state, crystallographic orientations, and void microstructure in intergranular coalescence. This second criterion is also assessed through comparisons to numerical limit analysis for an FCC crystal lattice. The agreement is very good in the case of coalescence by internal necking and the trends displayed by coalescence under combined tension and shear are captured correctly. Some implications of the model on the competition between transgranular and intergranular ductile fracture are discussed. Finally, by combining this model with an existing criterion for void growth at grain boundaries, a multi-surface yield function relevant to intergranular ductile fracture is obtained and compared to unit-cell simulations.

An uncoupled ductile fracture model considering void shape change and necking coalescence

In order to accurately describe the fracture behaviour of ductile metals in a wide range of stress states, based on the mechanism of void nucleation, growth, and coalescence during plastic deformation, an uncoupled isotropic ductile fracture model was established, which considered the combination of void shape change and void coalescence through internal necking. Firstly, the effect of the volume and shape changes on the fracture behaviour of the material was summarized and discussed, and the mechanism of interpore dimple shrinkage and consolidation during the void coalescence was studied. Secondly, the parameters of the proposed model are studied in detail to verify the flexibility of the new model. Then, the proposed ductile fracture model was used to construct the 3D fracture locus of AA 2024-T351, AISI 1045, Q460, Mg-Al-Zn-RE, and Ti-6Al-4V materials, and the predicted fracture strain values under corresponding stress states were compared with the experimental results of previous scholars. The effectiveness of the proposed new model is verified. Finally, to demonstrate the superiority of the proposed new model, the predicted results were compared with those of the widely used DF2016, MMC, and Hu models. The prediction results show that the new model has the highest precision compared to the other three. The void shape change and the void coalescence through internal necking play an essential role in the ductile metal fracture process, which should be fully considered.

Mechanism and Control Scheme of Central Defects in Cross Wedge Rolling of Railway Vehicle Axles

Faced with a great demand for railway axles, the cross wedge rolling (CWR) process has the advantages of high efficiency and material saving, and good forming quality of axles is significant for railway transportation safety. The stress inside the railway axle of CWR was analyzed by the finite element method. It was found that the center of the rolled piece is subjected to tensile stress in transverse and axial directions and compressive stress in radial direction, making it more prone to defects. By simulating the evolution of micro voids in the center of the CWR piece, it was found that the presence of voids makes the strain around them significantly large and concentrated and the material between the voids deforms intensely. When voids expand relative to the rolled piece and the internal necking between voids is significant, void coalescence is easy to occur, and central defects are formed. The influence of process parameters on void evolution was analyzed. The scheme of detaching die was proposed to avoid central defects of the CWR piece and the optimal parameter conditions of CWR of railway axles were determined, which proved that the quality of railway axles formed with optimized parameters meets the technical requirements of railway vehicle axles.

Yield surface for void growth and coalescence of porous anisotropic materials under axisymmetric loading

Ductile fracture in metallic alloys occurs by growth and coalescence of cavities. Growth, also referred to as homogeneous yielding, refers to rather diffuse plasticity around cavities, while coalescence, also termed as inhomogenous yielding, corresponds to the localization of plasticity along some planes or directions. Coalescence can develop in various patterns; three coalescence modes have been observed experimentally: internal necking, coalescence in columns and void sheeting. Plastic anisotropy of the material is known to have a significant effect on both homogeneous and inhomogeneous yielding. Therefore, in the present study, yield criteria accounting for the transition from homogeneous yielding to inhomogeneous yielding modes in anisotropic porous materials are obtained using kinematic limit analysis on a cylindrical unit-cell with a coaxial cylindrical cavity. Two types of plastic anisotropy are considered: Hill (1948) plasticity and crystal plasticity. The proposed analytical yield criteria are compared to numerical limit analysis computations and are found to qualitatively agree with simulations. In particular, plastic anisotropy, void shape effects and their coupling are well captured, especially regarding yield stresses and deformation modes. Finally, an homogenized model for Hill porous materials is obtained by supplementing evolution laws for microstructural parameters (void aspect ratio and ligament size ratios) derived from sequential limit analysis. Proposed evolution laws are then discussed in the light of numerical results and experimental evidence.

On the role of shape and distribution of secondary voids in the mechanism of coalescence

Ductile fracture of metals at room temperature has been studied extensively. Since localization of plastic flow is often viewed as a precursor to ductile fracture, several computational studies analyzing the influence of stress triaxiality and the Lode parameter on the critical strain for onset of localization have been reported. The implications of strain localization in materials containing randomly distributed voids for the mode of failure, that is, internal necking or shear banding have also been discussed. Apart from the influence of stress state, it is also of interest to model the effect of fine-scale microstructural attributes on the mode of failure and, hence, on material’s ductility. Therefore, in the present work, the mechanism of ductile fracture of an elastic–plastic solid containing two populations of pre-existing voids is modeled numerically. In the undeformed state, all the voids belonging to each family are assumed to have identical shape and size. Two different periodic spatial arrangements of primary voids are considered such that, in the absence of secondary voids, one of the configurations leads to internal necking whereas the other one exhibits a tendency for shear localization as the mode of coalescence for the larger voids. Small size secondary voids are then introduced explicitly and the role of shape and distribution of these sub-microscopic voids in the mechanism of coalescence and final failure is illustrated under plane strain as well as axisymmetric tension with a superposed hydrostatic stress. Depending on the microstructural attributes of the secondary voids, the mode of primary void coalescence is observed to shift from internal necking to void-sheeting and vice-versa. The implications of this shift in the mode of void coalescence on mesoscopic ductility is examined and a comparison of numerical predictions is made with the available experimental results.

On the micromechanics of voids in nanotwinned materials

This work investigates internal damage by void growth in NT microstructures via crystal plasticity. The framework incorporates length-scale effects and explicitly models twin boundary migration. Using finite-deformation, plane strain finite element calculations of porous unit cells, we analyze the roles of twin size, plastic anisotropy and twin boundary migration on void evolution over a range of controlled biaxial stress states. The simulations provide insights into crystallographic aspects of porosity growth in NT microstructures. Emphasis is placed on correlating the micromechanics of failure by internal necking. Irrespective of the level of crystallographic plastic anisotropy, twin boundary migration effectively shields the void growth process, delaying the porosity evolution compared to non-twinned microstructures. It is found that crystallographic plastic anisotropy causes kink band instability that can affect void growth. Coupled with twin boundary mobility, twin size and crystallographic plastic anisotropy create a rich landscape of failure characteristics as a function of the stress state.

On the micromechanics of void mediated failure in HCP crystals

Three-dimensional voided cell calculations are carried out to investigate the effect of initially flat voids on the effective behavior of hexagonal crystals. The crystalline matrix is described using a set of constitutive relations that account for slip and twinning. In some materials, the limiting case of penny-shaped cracks better represents incipient void formation. Cell boundary conditions are such that internal necking of the intervoid ligament or void impingement are the only possible failure modes. The overall behavior of the cell is analyzed under various stress states and two loading orientations, with special attention given to the evolution of damage-relevant microstructural variables. Particular emphasis is placed on crystallographic aspects of void growth and coalescence, which often prove difficult to represent using phenomenological anisotropic plasticity models.

A numerical study of a size-dependent finite-element based unit cell with primary and secondary voids

Aluminium alloys contain various types of intermetallic particles with different sizes, such as constituent particles and dispersoids. The main mechanism of ductile fracture in these materials is assumed to be nucleation of voids around the constituent particles, which grow during plastic deformation and eventually coalesce, resulting in material failure. The role of the dispersoids is less certain, but they are assumed to contribute in the last stages of the ductile fracture process. While the constituent particles are in the range of a couple of microns, the size of dispersoids is normally one order of magnitude smaller. To disclose the possible effects of the dispersoids on the ductile fracture process in aluminium alloys, this paper presents a numerical study of a finite-element based unit cell, which consists of a single spherical void embedded in a matrix material represented by a porous plasticity model with void size effects. Accordingly, the single, primary void of the unit cell is assumed to have nucleated on a constituent particle, whereas the matrix porosity is assumed to account for secondary, smaller voids nucleated on dispersoids. The effects of the intrinsic length scale of the matrix material on the void growth and coalescence are studied for a range of stress states, while the initial primary and secondary void volume fractions are kept constant. The secondary voids have a substantial effect on the behaviour of the unit cell when their size is large compared to the intrinsic material length scale, but they were not found to influence the growth of the primary void. Instead, the growth of the secondary voids promotes strain softening and influences the coalescence process of the primary voids, which gradually changes mode from internal necking to loss of load-carrying capacity of the inter-void ligament.

On the sensitivity of fracture mechanism to stress concentration configuration in a two-step quenching and partitioning steel

The effect of stress concentration geometry on the fracture behavior and fracture mechanisms of a two-step quenching and partitioning steel consisting of tempered martensite and retained austenite (RA) is investigated using double edge notched tension (DENT) specimens with different notch root radii. The initial notch opening is varied from a fatigue pre-crack to 15 mm radius. The fracture surface consists in flat triangular regions and slant zones. The fracture mode evolves from intergranular brittle fracture to ductile fracture with dimples. The ductile fracture process further changes from internal necking coalescence to a void sheet shear type coalescence when increasing the notch root radius. A similar transition of the fracture behavior is observed along the ligament of the DENT specimen. This change in the fracture mode results from a decrease of the stress triaxiality. The essential work of fracture is equal to $$40\hbox { kJ/m}^{{2}}$$. This is small when looking at the large value of the product of ultimate tensile strength and total elongation ($$\sim 25, 000\, \hbox {MPa}{\cdot }$$%). The small fracture toughness is ascribed to an intergranular fracture mode, which results from the distribution of blocky RA islands along the boundaries of martensitic packets and from the large stress concentration near the notch tip. A proper control of both the amount and morphology of RA during microstructure design is thus essential to generate the best compromise between tensile properties and fracture toughness, and to avoid the relatively low toughness and more brittle failure observed when stress triaxiality increases.

A nonlocal approach of ductile failure incorporating void growth, internal necking, and shear dominated coalescence mechanisms

An advanced modeling framework is developed for predicting the failure of ductile materials relying on micromechanics, physical ingredients, and robust numerical methods. The approach is based on a hyperelastic finite strain multi-surface constitutive model with multiple nonlocal variables. The three distinct nonlocal solutions for the expansion of voids embedded in an elastoplastic matrix are considered: a void growth phase governed by the Gurson–Tvergaard–Needleman yield surface, a void necking coalescence phase governed by a heuristic extension of the Thomason yield surface based on the maximum principal stress, and a competing void shearing coalescence phase triggered by the maximum shear stress. The first solution considers the diffused plastic deformation around the voids while the last two solutions correspond to a state of plastic localization between neighboring voids. This combination captures the Lode variable and shear effects, which play important roles in dictating the damage evolution rates. The implicit nonlocal formulation with multiple nonlocal variables, including the volumetric and deviatoric parts of the plastic strain, and the mean equivalent plastic strain of the matrix, regularizes the problem of the loss of solution uniqueness when material softening occurs whatever the localization mechanism. The predictive capability of the proposed model is demonstrated through different numerical simulations in which complex failure patterns such as slant and cup-cone of respectively plane strain and axisymmetric samples under tensile loading conditions develop.

Deep Crustal Flow Within Postorogenic Metamorphic Core Complexes: Insights From the Southern Western Gneiss Region of Norway

AbstractViscous crustal flow can exhume once deeply buried rocks in postorogenic metamorphic core complexes (MCCs). While migmatite domes record the flow dynamics of anatectic crust, the mechanics and kinematics of solid‐state flow in the deep crust are poorly constrained. To address this issue, we studied a deeply eroded and particularly well‐exposed MCC in the southern Western Gneiss Region of Norway. The Gulen MCC formed during Devonian transtensional collapse of the Caledonian orogeny in the footwall of the Nordfjord‐Sogn detachment zone. We developed a semiquantitative mapping scheme for ductile strain to constrain micro‐ to megascale processes, which brought eclogite‐bearing crust from the orogenic root into direct contact with Devonian supradetachment basins. The Gulen MCC comprises different structural levels with distinct metamorphic evolutions. In the high‐grade core, amphibolite‐facies structures record fluid‐controlled eclogite retrogression and coaxial flow involving vast extension‐perpendicular shortening. Detachment mylonites formed during ductile‐to‐brittle noncoaxial deformation and wrap around the core. We present a sequential 3‐D reconstruction of MCC formation. In the detachment zone, the combined effects of simple shearing, incision/excision, and erosion thinned the upper crust. Internal necking of the ductile crust was compensated by extension‐perpendicular shortening within the deep crust and resulted in differential folding of distinct crustal levels. We identify this differential folding as the main mechanism that can redistribute material within solid‐state MCCs. Our interpretation suggests a continuum of processes from migmatite‐cored to solid‐state MCCs and has implications for postorogenic exhumation of (ultra‐)high‐pressure rocks.

Hydrogen informed Gurson model for hydrogen embrittlement simulation

Hydrogen-microvoid interactions were studied via unit cell analyses with different hydrogen concentrations. The absolute failure strain decreases with hydrogen concentration, but the failure loci were found to follow the same trend dependent only on stress triaxiality, in other words, the effects of geometric constraint and hydrogen on failure are decoupled. Guided by the decoupling principle, a hydrogen informed Gurson model is proposed. This model is the first practical hydrogen embrittlement simulation tool based on the hydrogen enhanced localized plasticity (HELP) mechanism. It introduces only one additional hydrogen related parameter into the Gurson model and is able to capture hydrogen enhanced internal necking failure of microvoids with accuracy; its parameter calibration procedure is straightforward and cost efficient for engineering purpose.

On the fracture behavior and toughness of TA15 titanium alloy with tri-modal microstructure

In this work, the micro-scale damage development and crack propagation features of TA15 titanium alloy with tri-modal microstructure consisting of equiaxed α (αp), lamellar α (αl) and β transformed matrix α (βt) were firstly investigated. On that basis, the dependences of fracture toughness of tri-modal microstructure on microstructural parameters were discussed. The results indicate that in the damage and fracture of tri-modal microstructure, voids mainly nucleate and grow up at the triple junctions and the interfaces of αp/αp, αp/βt and single-αl/βt due to the micro-scale strain incompatibility at these locations. By internal necking or local shearing connection with these voids, the main crack achieves an increment. It leads to two main categories of crack paths, i.e., the connection-type path caused by the connection behavior and the natural-extension-type path progressing through voids after the connection with voids. The connection-type path exerts greater influence on the fracture toughness than the natural-extension-type path. Moreover, as for the connection-type path, it can also be classified into two sub-categories, i.e., the internal necking connection-type path and the local shearing connection-type path. As far as the energy consumption is considered, more internal necking connection-type path is favorable for improving the fracture toughness, while more local shearing connection-type path is detrimental. Besides, the above void evolution and crack propagation behavior will affect the tortuosity of crack path to some extent, which will also change the overall energy consumption and final fracture toughness. As for the dependence of fracture toughness on microstructural parameters, it is found that with the increase of αp content, the proportion of low-energy-consumed crack paths (local shearing across αp and colony-αl) increase and the crack path tortuosity decreases, both of which reduce the energy consumption and result in the continuous reduction of fracture toughness. As the αl content increases, the disorderly distributed single-αl significantly increases at lower αl content, which improves the crack path tortuosity and energy consumption. However, at a higher level of αl content, single-αl decreases dramatically and the content of colony-αl becomes high, which reduces the crack path tortuosity and energy consumption remarkably. So, the fracture toughness first increases and then decreases with the increase of αl content.

Four-dimensional observation of ductile fracture in sintered iron using synchrotron X-ray laminography

ABSTRACTSynchrotron X-ray laminography was used to examine the time-dependent evolution of the three-dimensional (3D) morphology of micropores in sintered iron during the tensile test. 3D snapshots showed that the networked open pores grow wider than 20 µm along the tensile direction, resulting in the internal necking of the specimen. Subsequently, these pores initiated the cracks perpendicular to the tensile direction by coalescing with the surrounding pre-existing microvoids or with the secondary-generated voids immediately before fracture. Topological analysis of the barycentric positions of these microvoids showed that they form the two-dimensional networks within the ∼20 µm of radius area. These observations strongly indicate that the microvoid coalescence could occur on shear planes formed close to the enlarged open pores or between closed pores by strain accumulation and play an important role in the crack initiation.

A unified criterion for void growth and coalescence under combined tension and shear

An analytical micromechanics-based yield criterion is developed to describe both void growth and coalescence under combined tension and shear, with smooth transition between growth and coalescence, thus its name unified. The model is obtained by limit analysis over a cylindrical elementary cell embedding a coaxial cylindrical void of finite height. The velocity field employed is an extended counterpart of the discontinuous, yet kinematically admissible trial field utilized in a recent work. Plasticity in the deformable matrix is modeled using rate-independent J2 flow theory, and the effective dissipation function is calculated by exact as well as approximate integration techniques, the latter generating a simpler flow potential. The model is aimed to predict void growth as well as coalescence by internal necking or shearing. The complete yield surface, being function of normal as well as shear stresses, exhibits curved and planar parts signifying void coalescence. The transition between the curved and planar parts is cornerless. The analytical predictions are compared to results of FEM single-step cell-model calculations of limit analysis executed on an identical geometry exposed to quasi-periodic boundary conditions.

Void coalescence in porous ductile solids containing two populations of cavities

Abstract A model of coalescence by internal necking of primary voids is developed which accounts for the presence of a second population of cavities. The derivation is based on a limit-analysis of a cylindrical cell containing a mesoscopic void and subjected to boundary conditions describing the kinematics of coalescence. The second population is accounted locally in the matrix surrounding the mesoscopic void through the microscopic potential of Michel and Suquet (1992) for spherical voids. The macroscopic criterion obtained is assessed through comparison of its predictions with the results of micromechanical finite element simulations on the same cell. A good agreement between model predictions and numerical results is found on the limit-load promoting coalescence.

Combined effects of triaxiality, Lode parameter and shear stress on void growth and coalescence

Shear stress has been recognized as an important parameter, in addition to stress triaxiality and Lode parameter, that characterizes the stress state and influences the ductile fracture process. Micromechanical analyses are performed on a cubic unit cell with a centrally embedded spherical void in this paper. An effective method for controlling triaxiality, Lode parameter and shear stress component throughout the loading history is provided. The combined normal and shear stress is applied to investigate the effects of shear stress as well as the triaxiality and Lode parameter on the void growth and coalescence as well as material response. The results show that increasing triaxiality, shear stress and Lode parameter decrease the loading carrying capacity of the ductile solid and the rate of void growth at high and intermediate triaxialities. The unit cell model is then used to analyze two different void coalescence mechanisms, including internal necking at high and intermediate triaxialities and shearing at low triaxialities. The increasing weight of shear stress component facilitates the internal necking at high and intermediate triaxialities. At low triaxialities, the shear stress benefits to the void rotation, elongation, coalescence and reducing the distance of ligament between adjacent voids. Comparison of the state variables at the onset of void coalescence between the numerical modeling and the analytical criteria is performed. The results indicate that improvement to the current criteria is needed to obtain more accurate predictions of the onset of void coalescence under combined normal and shear stress over a wide range of triaxialities.

Hydrogen-microvoid interactions at continuum scale

Experimentally, it has been shown that hydrogen can either enhance the internal necking failure or induce internal shearing failure of microvoids. In this study, the numerical investigation of hydrogen-microvoid interactions under the framework of hydrogen enhanced localized plasticity mechanism reveals that the actual effect of hydrogen depends on the stress state as well as on the hydrogen trapping effect. Hydrogen enhanced internal necking failure is observed over the entire stress space at a low level of trapping effect. While such failure is still observed in the high triaxiality regime at a high level of trapping effect, hydrogen induced internal shearing failure is observed in the low triaxiality regime. A hydrogen induced internal shearing failure criterion is proposed, and the failure loci corresponding to the low and high levels of trapping effect are constructed. The hydrogen induced internal shearing failure locus is found to be approximately independent of stress triaxiality while the hydrogen enhanced internal necking failure locus maintains similar triaxiality dependency as in the absence of hydrogen. The loss of ductility, in terms of reduction in failure strain and dimple size, is more pronounced in the case of hydrogen induced internal shearing failure. Subsequent study of the Lode effect reveals that plane strain tension is the most critical case for hydrogen induced internal shearing failure while axisymmetric tension is the most critical for hydrogen enhanced internal necking failure.